- #1
jjou
- 64
- 0
(Problem 49 from practice GRE Math exam:) Up to isomorphism, how many additive abelian groups G of order 16 have the property that x + x + x + x = 0 for each x in G?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
The answer is (D) 3, but I don't understand what the problem is asking, really, and I don't know what strategy I should take for this problem. My first guess was to just construct some groups having this property, but I think, on the actual exam, this strategy would take too long.
I was able to construct two groups, but I can't figure out what the third is.
[tex]G_1=\{0, 1, x, x+1, x^2, x^2+1, x^2+x, x^2+x+1, x^3, x^3+1, x^3+x, x^3+x+1, x^3+x^2, x^3+x^2+1, x^3+x^2+x, x^3+x^2+x+1\}[/tex]
[tex]G_2=\{0, 1, 2, 3, x, x+1, x+2, x+3, 2x, 2x+1, 2x+2, 2x+3, 3x, 3x+1, 3x+2, 3x+3\}[/tex]
Does anyone know what the third group is or have a better method for solving this problem?
Any suggestions / insights would be appreciated!
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
The answer is (D) 3, but I don't understand what the problem is asking, really, and I don't know what strategy I should take for this problem. My first guess was to just construct some groups having this property, but I think, on the actual exam, this strategy would take too long.
I was able to construct two groups, but I can't figure out what the third is.
[tex]G_1=\{0, 1, x, x+1, x^2, x^2+1, x^2+x, x^2+x+1, x^3, x^3+1, x^3+x, x^3+x+1, x^3+x^2, x^3+x^2+1, x^3+x^2+x, x^3+x^2+x+1\}[/tex]
[tex]G_2=\{0, 1, 2, 3, x, x+1, x+2, x+3, 2x, 2x+1, 2x+2, 2x+3, 3x, 3x+1, 3x+2, 3x+3\}[/tex]
Does anyone know what the third group is or have a better method for solving this problem?
Any suggestions / insights would be appreciated!